khalamar’s answer is correct and probably the simplest one so far.
But we can do even simpler, in the same spirit.
Start with a fraction:
> (a / b) / (c / d)
Multiply both sides of the fraction by (d), you get:
> (d * a / b) / ( c )
Multiply both sides again by (b) , you get:
> (d * a) / (c * b)
Notice that this is equivalent to the flip you were mentioning.
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